# On generalized Stieltjes functions

## TL;DR

This paper characterizes generalized Stieltjes functions of order mbda>0 by their derivatives' complete monotonicity, extending Sokal's result, and provides a measure-based characterization for related functions.

## Contribution

It offers a new characterization of generalized Stieltjes functions and their derivatives' monotonicity properties, expanding the theoretical understanding of these functions.

## Key findings

- Characterization of generalized Stieltjes functions via complete monotonicity of derivatives.
- Extension of Sokal's result to broader classes of functions.
- Measure-based criteria for functions with derivatives exhibiting partial complete monotonicity.

## Abstract

It is shown that a function $f$ is a generalized Stieltjes function of order $\lambda>0$ if and only if $x^{1-\lambda}(x^{\lambda-1+k}f(x))^{(k)}$ is completely monotonic for all $k\geq 0$, thereby complementing a result due to Sokal. Furthermore, a characterization of those completely monotonic functions $f$ for which $x^{1-\lambda}(x^{\lambda-1+k}f(x))^{(k)}$ is completely monotonic for all $k\leq n$ is obtained in terms of properties of the representing measure of $f$.